Lemma 6.36 (See [1]}.) 
 Referring to Notation REF ,
with respect to the basis in Lemma REF ,
the matrices representing \(A\)  and \(A^*\)  are
\(A & :\begin{pmatrix}a_0 & b_0 & & & & \text{\bf 0} \\c_1 & a_1 & b_1 \\& c_2 & \cdot & \cdot \\& & \cdot & \cdot & \cdot \\& & & \cdot & \cdot & b_{d-1} \\\text{\bf 0} & & & & c_d & a_d\end{pmatrix},&A^* & :\text{\rm diag}(\theta ^*_0, \theta ^*_1, \ldots , \theta ^*_d).\) 

